On the effect of rotation on the life-span of analytic solutions to the $3D$ inviscid primitive equations

Ghoul, Tej-Eddine; Ibrahim, Slim; Lin, Quyuan; Titi, Edriss S.

doi:10.48550/arxiv.2010.01740

Cited by 4 publications

(5 citation statements)

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“…In the last few years, some developments concerning the global well-posedness to the anisotropic primitive equations were also made, see Cao-Titi [5] and Cao-Li-Titi [3,4,6,7,8], which in particular imply that the primitive equations with only horizontal viscosities are globally well-posed as long as one still has either horizontal or vertical diffusivities, see also [15] and [30]. Notably, different from the primitive equations with either full viscosity or only horizontal viscosity, the inviscid primitive equations may develop finite time singularities, see Cao et al [11], Wong [55], Ghoul et al [20] and Ibrahim et al [31].…”

Section: Introductionmentioning

confidence: 99%

The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

Titi

Yuan

2022

Journal of Differential Equations

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Section: Introductionmentioning

confidence: 99%

The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

Titi

Yuan

2022

Journal of Differential Equations

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“…Indeed, the linear and nonlinear ill-posedness in any Sobolev space have been established in [32] and in [19], respectively. On the other hand, by assuming either real analyticity or some special structures (local Rayleigh condition) on the initial data, one is able to establish the local well-posedness, see [2,3,17,18,24,25,29]. Moreover, it was proven that smooth solutions to the inviscid PEs, in the absence of rotation, can develop singularities in finite time.…”

Section: Introductionmentioning

confidence: 99%

“…Note that (10) without pressure is (17) without viscosity. Interestingly, the blow-up dynamics of Theorem 1.1 and that of [10] are genuinely different, due to the absence of a Dirichlet boundary condition for (10) allowing for a blow-up at the boundary, and to the absence of pressure and confinement z ∈ [0, 1] in (17) allowing for the transverse spatial scale ν to grow to infinity so to maintain the divergence free condition.…”

Section: Introductionmentioning

confidence: 99%

Stable Singularity Formation for the Inviscid Primitive Equations

Collot,

Ibrahim,

Lin

2021

Preprint

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The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations.

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“…The results in [4-9, 12, 31] indicate that the horizontal viscosity has a crucial effect on the global well-posedness of the primitive equations. The effect of the rotation on the life-span of solutions to the inviscid primitive equations was studied in [19].…”

Section: Introductionmentioning

confidence: 99%

Global well-posedness of $z$-weak solutions to the primitive equations without vertical diffusivity

Li,

Yuan

2021

Preprint

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In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of z-weak solution is established for any such initial datum that itself and its vertical derivative belong to L 2 . This not only extends the results in [8] from the spatially periodic case to general cylindrical domains but also weakens the regularity assumptions on the initial data which are required to be H 2 there.

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On the effect of rotation on the life-span of analytic solutions to the $3D$ inviscid primitive equations

Cited by 4 publications

References 0 publications

The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

Stable Singularity Formation for the Inviscid Primitive Equations

Global well-posedness of $z$-weak solutions to the primitive equations without vertical diffusivity

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